Unbounded order convergence in dual spaces

A net (x(alpha)) in a vector lattice X is said to be unbounded order convergent (or uo-convergent, for short) to x is an element of X if the net (vertical bar x(alpha) - x vertical bar boolean AND y)converges to 0 in order for all y is an element of X+. In this paper, we study unbounded order convergence in dual spaces of Banach lattices. Let X be a Banach lattice. We prove that every norm bounded uo-convergent net in X* is w*-convergent if X has order continuous norm, and that every w*-convergent net in X* is no-convergent if X is atomic with order continuous norm. We also characterize among sigma-order complete Banach lattices the spaces in whose dual space every simultaneously uo- and w*-convergent sequence converges weakly/in norm. (C) 2014 Elsevier Inc. All rights reserved.